Estimate the area under the graph of f(x) = 1+ x². from x= -1 to x=2 using three rectangles and right endpoints. Then improve the estimate by using six rectangles.

Question

1. Repeat using left endpoints.
2. Repeat using midpoints.
3. From the sketches, which is the best estimate?

Answer 1

0) Right endpoints of x = {-1,0,1,2}, y = {2,1,2,5} yields R3 = [1+2+5]•(Δx= 1.0) = 8.0. For 6 intervals, x = {-1,-0.5,0,0.5,1,1.5,2}, y = {2,1.25,1,1.25,2,3.25,5} and R6 = {sum of all but 1st}•(0.5) = 6.875.

1) Left endpoints of the above yield L3 = 5.0 and L6 = 5.375.

2) Midpoints for 3 intervals are x = {-0.5,0.5,1.5}, y = {1.25,1.25,3.25} and M3 = 5.75. For 6 intervals x = {-0.75,-0.25,0.25,0.75,1.25,1.75}, y = {1.5625,1.0625,1.0625,1.5625,2.5625,4.0625} and M6 = 5.9375.

3) Given the curvature of the parabola with vertex at x = 0, the midpoint result M6 has to be the best estimate.